The relative GAGA Theorem and an application to the analytic mapping stacks

We prove a relative GAGA theorem for perfect and pseudo-coherent complexes in non-archimedean analytic geometry, allowing bases given by Fredholm analytic rings, including those associated from affinoid perfectoid spaces. This answers a question raised in \cite{heuer2024padicnonabelianhodgetheory}. As an application, we show that for a proper scheme $X$ and an Artin stack $Y$ with suitable conditions, the analytification of the algebraic mapping stack $\mathrm{M}\mathrm{a}\mathrm{p}(X,Y)$ agrees with the intrinsic analytic mapping stack $\mathrm{M}\mathrm{a}\mathrm{p}(X^{\mathrm{a}\mathrm{n}},Y^{\mathrm{a}\mathrm{n}})$.